BIFURCATION AND CHAOS IN COUPLED BVP OSCILLATORS

2004 ◽  
Vol 14 (04) ◽  
pp. 1305-1324 ◽  
Author(s):  
TETSUSHI UETA ◽  
HISAYO MIYAZAKI ◽  
TAKUJI KOUSAKA ◽  
HIROSHI KAWAKAMI
Keyword(s):  
Autonomous System ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Nonlinear Phenomena ◽  
Van Der Pol ◽  
Bifurcation And Chaos ◽  
Experimental Laboratory ◽  
Chaotic Parameter ◽  
Parameter Values ◽  
The Mathematical Model

Bonhöffer–van der Pol(BVP) oscillator is a classic model exhibiting typical nonlinear phenomena in the planar autonomous system. This paper gives an analysis of equilibria, periodic solutions, strange attractors of two BVP oscillators coupled by a resister. When an oscillator is fixed its parameter values in nonoscillatory region and the others in oscillatory region, create the double scroll attractor due to the coupling. Bifurcation diagrams are obtained numerically from the mathematical model and chaotic parameter regions are clarified. We also confirm the existence of period-doubling cascades and chaotic attractors in the experimental laboratory.

ZERO-CROSS INSTANTANEOUS STATE SETTING FOR CONTROL OF A BIFURCATING H-BRIDGE INVERTER

2007 ◽  
Vol 17 (10) ◽  
pp. 3571-3575 ◽  
Author(s):  
SATOSHI AKATSU ◽  
HIROYUKI TORIKAI ◽  
TOSHIMICHI SAITO
Keyword(s):  
Pulse Width Modulation ◽  
Control Method ◽  
Period Doubling ◽  
Current Mode ◽  
Nonlinear Phenomena ◽  
Unstable Periodic Orbits ◽  
Bifurcation And Chaos ◽  
Zero Crossing ◽  
Modulation Signal ◽  
A Current

This paper studies stabilization of low-period unstable periodic orbits (UPOs) in a simplified model of a current mode H-bridge inverter. The switching of the inverter is controlled by pulse-width modulation signal depending on the sampled inductor current. The inverter can exhibit rich nonlinear phenomena including period doubling bifurcation and chaos. Our control method is realized by instantaneous opening of inductor at a zero-crossing moment of an objective UPO and can stabilize the UPO instantaneously as far as the UPO crosses zero in principle. Typical system operations can be confirmed by numerical experiments.

scholarly journals Colpitts Chaotic Oscillator Coupling with a Generalized Memristor

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Ling Lu ◽  
Changdi Li ◽  
Zicheng Zhao ◽  
Bocheng Bao ◽  
Quan Xu
Keyword(s):  
Mathematical Model ◽  
Equilibrium Point ◽  
Fourth Order ◽  
Chaotic Attractors ◽  
Chaotic Oscillator ◽  
Nonlinear Phenomena ◽  
Unique Equilibrium ◽  
First Order ◽  
The Mathematical Model ◽  
Order Parallel

By introducing a generalized memristor into a fourth-order Colpitts chaotic oscillator, a new memristive Colpitts chaotic oscillator is proposed in this paper. The generalized memristor is equivalent to a diode bridge cascaded with a first-order parallel RC filter. Chaotic attractors of the oscillator are numerically revealed from the mathematical model and experimentally captured from the physical circuit. The dynamics of the memristive Colpitts chaotic oscillator is investigated both theoretically and numerically, from which it can be found that the oscillator has a unique equilibrium point and displays complex nonlinear phenomena.

Boundary-Crisis-Induced Complex Bursting Patterns in a Forced Cubic Map

2017 ◽  
Vol 27 (04) ◽  
pp. 1750051 ◽  
Author(s):  
Xiujing Han ◽  
Chun Zhang ◽  
Yue Yu ◽  
Qinsheng Bi
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Period 2 ◽  
Boundary Crisis ◽  
Underlying Mechanisms ◽  
Parameter Values ◽  
Point Type ◽  
Period Doubling Bifurcations

This paper reports novel routes to complex bursting patterns based on a forced cubic map, in which boundary-crisis-induced novel bursting patterns are investigated. Typically, the cubic map exhibits stable upper and lower branches of fixed points, which may evolve into chaos in opposite parameter directions by a cascade of period-doubling bifurcations. We show that the chaotic attractors on the stable branches may suddenly disappear by boundary crisis, thus leading to fast transitions from chaos to other attractors and giving rise to switchings between the stable branches of solutions of the cubic map. In particular, the attractors that the trajectory switches to by boundary crisis can be fixed points, periodic orbits and chaos, dependent on parameter values of the cubic map, and this helps us to reveal three general types of boundary-crisis-induced bursting, i.e. bursting of chaos-point type, bursting of chaos-cycle type and bursting of chaos-chaos type. Moreover, each bursting type may contain various bursting patterns. For bursting of chaos-cycle type, we see rich bursting patterns, e.g. chaos-period-2 bursting, chaos-period-4 bursting, chaos-period-8 bursting, etc. Our results enrich the possible routes to complex bursting patterns as well as the underlying mechanisms of complex bursting patterns.

scholarly journals Global bifurcation analysis of Topp system

2019 ◽  
pp. 244-250
Author(s):  
Valery A. Gaiko ◽  
Henk W. Broer ◽  
Alef E. Sterk
Keyword(s):  
Limit Cycles ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Homoclinic Orbits ◽  
Chaotic Attractors ◽  
Dimensional Model ◽  
Saddle Node Bifurcation ◽  
3 Dimensional ◽  
Suitable Parameter ◽  
Parameter Values

In this paper, we study the 3-dimensional Topp model for the dynamics of diabetes. First, we reduce the model to a planar quartic system. In particular, studying global bifurcations, we prove that such a system can have at most two limit cycles. Next, we study the dynamics of the full 3-dimensional model. We show that for suitable parameter values an equilibrium bifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests that near this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arise through period doubling cascades of limit cycles.

BIFURCATION STRUCTURE OF THE DRIVEN VAN DER POL OSCILLATOR

1993 ◽  
Vol 03 (06) ◽  
pp. 1529-1555 ◽  
Author(s):  
R. METTIN ◽  
U. PARLITZ ◽  
W. LAUTERBORN
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Van Der Pol Oscillator ◽  
Chaotic Attractors ◽  
Transitional Region ◽  
Driving Frequency ◽  
Van Der Pol ◽  
Cycle Frequency ◽  
Dimensional Parameter ◽  
Periodically Driven

The bifurcation set in the three-dimensional parameter space of the periodically driven van der Pol oscillator has been investigated by continuation of local bifurcation curves. The two regions in which the driving frequency ω is greater or less than the limit cycle frequency ω0 of the nondriven oscillator are considered separately. For the case ω > ω0, the subharmonic region, the extent and location of the largest Arnol'd tongues are shown, as well as the period-doubling cascades and chaotic attractors that appear within most of them. Special attention is paid to the pattern of the bifurcation curves in the transitional region between low and large dampings that is difficult to approach analytically. In the case ω < ω0, the ultraharmonic region, a recurrent pattern of the bifurcation curves is found for small values of the damping d. At medium damping the structure of the bifurcation curves becomes involved. Period-doubling sequences and chaotic attractors occur.

scholarly journals Experiment on Bifurcation and Chaos in Coupled Anisochronous Self-Excited Systems: Case of Two Coupled van der Pol-Duffing Oscillators

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
J. Kengne ◽  
F. Kenmogne ◽  
V. Kamdoum Tamba
Keyword(s):  
Analog Circuit ◽  
Period Doubling ◽  
Van Der Pol ◽  
Bifurcation And Chaos ◽  
Small Oscillations ◽  
Driven Systems ◽  
Electrical Systems ◽  
The Difference ◽  
Theoretical Analyses ◽  
Analog Circuit Implementation

The analog circuit implementation and the experimental bifurcation analysis of coupled anisochronous self-driven systems modelled by two mutually coupled van der Pol-Duffing oscillators are considered. The coupling between the two oscillators is set in a symmetrical way that linearly depends on the difference of their velocities (i.e., dissipative coupling). Interest in this problem does not decrease because of its significance and possible application in the analysis of different, biological, chemical, and electrical systems (e.g., coupled van der Pol-Duffing electrical system). Regions of quenching behavior as well as conditions for the appearance of Hopf bifurcations are analytically defined. The scenarios/routes to chaos are studied with particular emphasis on the effects of cubic nonlinearity (that is responsible for anisochronism of small oscillations). When monitoring the control parameter, various striking dynamic behaviors are found including period-doubling, symmetry-breaking, multistability, and chaos. An appropriate electronic circuit describing the coupled oscillator is designed and used for the investigations. Experimental results that are consistent with results from theoretical analyses are presented and convincingly show quenching phenomenon as well as bifurcation and chaos.

Dynamical Effects of Offset Terms on a Modified Chua’s Oscillator and Its Circuit Implementation

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Léandre Kamdjeu Kengne ◽  
Karthikeyan Rajagopal ◽  
Nestor Tsafack ◽  
Paul Didier Kamdem Kuate ◽  
Balamurali Ramakrishnan ◽  
...  
Keyword(s):  
Initial Conditions ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Phenomena ◽  
Hidden Attractor ◽  
Chua’S Oscillator ◽  
Kirchhoff’S Laws ◽  
Attraction Basins ◽  
The Mathematical Model

This paper addresses the effects of offset terms on the dynamics of a modified Chua’s oscillator. The mathematical model is derived using Kirchhoff’s laws. The model is analyzed with the help of the maximal Lyapunov exponent, bifurcation diagrams, phase portraits, and basins of attraction. The investigations show that the offset terms break the symmetry of the system, generating more complex nonlinear phenomena like coexisting asymmetric bifurcations, coexisting asymmetric attractors, asymmetric double-scroll chaotic attractors and asymmetric attraction basins. Also, a hidden attractor (period-1 limit cycle) is found when varying the initial conditions. More interestingly, this latter attractor coexists with all other self-excited ones. A microcontroller-based implementation of the circuit is carried out to verify the numerical investigations.

scholarly journals Bifurcation Analysis of a Van der Pol-Duffing Circuit with Parallel Resistor

2009 ◽  
Vol 2009 ◽  
pp. 1-26 ◽  
Author(s):  
Denis de Carvalho Braga ◽  
Luis Fernando Mello ◽  
Marcelo Messias
Keyword(s):  
Analytical Study ◽  
Complex Dynamics ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Van Der Pol ◽  
Codimension One ◽  
Local Bifurcations ◽  
Parameter Values ◽  
Small Periodic

We study the local codimension one, two, and three bifurcations which occur in a recently proposed Van der Pol-Duffing circuit (ADVP) with parallel resistor, which is an extension of the classical Chua's circuit with cubic nonlinearity. The ADVP system presents a very rich dynamical behavior, ranging from stable equilibrium points to periodic and even chaotic oscillations. Aiming to contribute to the understand of the complex dynamics of this new system we present an analytical study of its local bifurcations and give the corresponding bifurcation diagrams. A complete description of the regions in the parameter space for which multiple small periodic solutions arise through the Hopf bifurcations at the equilibria is given. Then, by studying the continuation of such periodic orbits, we numerically find a sequence of period doubling and symmetric homoclinic bifurcations which leads to the creation of strange attractors, for a given set of the parameter values.

Transitions Through Period Doubling Route to Chaos

1991 ◽  
Author(s):  
R. M. Evan-lwanowski ◽  
Chu-Ho Lu
Keyword(s):  
Period Doubling ◽  
Mirror Image ◽  
Flip Flop ◽  
Physical Systems ◽  
Starting Conditions ◽  
Spatial System ◽  
Stationary Bifurcation ◽  
Extreme Sensitivity ◽  
Route To Chaos ◽  
Parameter Values

Abstract The Duffing driven, damped, “softening” oscillator has been analyzed for transition through period doubling route to chaos. The forcing frequency and amplitude have been varied in time (constant sweep). The stationary 2T, 4T… chaos regions have been determined and used as the starting conditions for nonstationary regimes, consisting of the transition along the Ω(t)=Ω0±α2t,f=const., Ω-line, and along the E-line: Ω(t)=Ω0±α2t;f(t)=f0∓α2t. The results are new, revealing, puzzling and complex. The nonstationary penetration phenomena (delay, memory) has been observed for a single and two-control nonstationary parameters. The rate of penetrations tends to zero with increasing sweeps, delaying thus the nonstationary chaos relative to the stationary chaos by a constant value. A bifurcation discontinuity has been uncovered at the stationary 2T bifurcation: the 2T bifurcation discontinuity drops from the upper branches of (a, Ω) or (a, f) curves to their lower branches. The bifurcation drops occur at the different control parameter values from the response x(t) discontinuities. The stationary bifurcation discontinuities are annihilated in the nonstationary bifurcation cascade to chaos — they reside entirely on the upper or lower nonstationary branches. A puzzling drop (jump) of the chaotic bifurcation bands has been observed for reversed sweeps. Extreme sensitivity of the nonstationary bifurcations to the starting conditions manifests itself in the flip-flop (mirror image) phenomena. The knowledge of the bifurcations allows for accurate reconstruction of the spatial system itself. The results obtained may model mathematically a number of engineering and physical systems.

Some Nonlinear Effects of Magnetic Bearings

1999 ◽  
Author(s):  
Norbert Steinschaden ◽  
Helmut Springer
Keyword(s):  
Equations Of Motion ◽  
Period Doubling ◽  
Path Following ◽  
Nonlinear Effects ◽  
Magnetic Bearing ◽  
Operating Conditions ◽  
Active Magnetic Bearing ◽  
Nonlinear Phenomena ◽  
Amb System ◽  
Large Rotor

Abstract In order to get a better understanding of the dynamics of active magnetic bearing (AMB) systems under extreme operating conditions a simple, nonlinear model for a radial AMB system is investigated. Instead of the common way of linearizing the magnetic forces at the center position of the rotor with respect to rotor displacement and coil current, the fully nonlinear force to displacement and the force to current characteristics are used. The AMB system is excited by unbalance forces of the rotor. Especially for the case of large rotor eccentricities, causing large rotor displacements, the behaviour of the system is discussed. A path-following analysis of the equations of motion shows that for some combinations of parameters well-known nonlinear phenomena may occur, as, for example, symmetry breaking, period doubling and even regions of global instability can be observed.

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